Abstract

Let $\phi$ and ψ be regular quadratic forms over a field F of characteristic different from 2. We say that ψ is a quasisubform of $\phi$ if there is $a\in F^{*}$ such that $a\psi$ is a subform of $\phi$. Let L/F be an odd degree field extension. Assume that ψ_L is a quasisubform of ${\phi }_L$. A natural question is whether ψ is a quasisubform of $\phi$. We give a positive answer to this question in any of the following cases:

1. The 2-cohomological dimension of F equals 2.

2. $\text{dim\hspace{0.17em}}\phi -\text{dim\hspace{0.17em}}\psi \le 1,$ or $\text{dim\hspace{0.17em}}\phi -\text{dim\hspace{0.17em}}\psi =2$ and $\text{dim\hspace{0.17em}}\phi$ is even.

3. $\text{dim\hspace{0.17em}}\psi =3,$ and $\text{dim\hspace{0.17em}}\phi \le 6.$

4. $\text{dim\hspace{0.17em}}\psi =3,$ and $⟪a,b⟫I^2(F)=0,$ where $⟪a,b⟫$ is the Pfister form associated with ψ (the most difficult case).

摘要

让 $\phi$并且ψ是特征域F 上的规则二次型，其特征不同于 2。我们说ψ是$\phi$ 如果有 $\mathrm{一种}\in F^{*}$ 以至于 $\mathrm{一种}\psi$ 是一个子形式 $\phi$. 令L / F为奇数场扩展。假设ψ _L是${\phi }_{升}$. 一个自然的问题是ψ是否是$\phi$. 在以下任何一种情况下，我们都会对这个问题给出肯定的回答：

1. F的 2-上同调维数等于 2。

2. $\text{暗淡\hspace{0.17em}}\phi -\text{暗淡\hspace{0.17em}}\psi \le 1,$ 或者 $\text{暗淡\hspace{0.17em}}\phi -\text{暗淡\hspace{0.17em}}\psi =2$ 和 $\text{暗淡\hspace{0.17em}}\phi$ 甚至。

3. $\text{暗淡\hspace{0.17em}}\psi =3,$ 和 $\text{暗淡\hspace{0.17em}}\phi \le 6.$

4. $\text{暗淡\hspace{0.17em}}\psi =3,$ 和 $⟪\mathrm{一种},乙⟫{\mathrm{一世}}^2(F)=0,$ 在哪里 $⟪\mathrm{一种},乙⟫$是与ψ相关的 Pfister 形式（最困难的情况）。